BRST cohomology and phase space reduction in deformation quantization. (English) Zbl 0961.53046

The BRST cohomology in the framework of deformation quantization is defined. The starting point is a star-product \(*\) for a Poisson structure on a manifold \(M\), defined on the algebra \(C^\infty(M)[[\lambda]]\), and a Hamiltonian action of a Lie group \(G\) in \(M\). With this action a quantum momentum map \({\mathbf J}=\sum_{r=0}^\infty\lambda^r{\mathbf J}_r:M\rightarrow {\mathfrak g}^*[[\lambda]]\) is associated, for which \({\mathbf J}_0=J\) is a classical equivariant momentum map. The star product is taken to be quantum covariant, i.e., such that \(\langle{\mathbf J},\xi\rangle *\langle{\mathbf J},\eta\rangle - \langle{\mathbf J},\eta\rangle *\langle{\mathbf J},\xi\rangle= \text{i}\lambda \langle{\mathbf J},[\xi,\eta]\rangle\) for all \(\xi,\eta\) from the Lie algebra \({\mathfrak g}\) of \(G\). The underlying vector space for the quantum BRST algebra is the \({\mathbb{C}}[[\lambda]]\)-module \({\mathcal A}[[\lambda]]\) of formal power series with values in the classical super Poisson algebra \({\mathcal A}=\bigwedge{\mathfrak g}^*\otimes\bigwedge{\mathfrak g}\otimes C^\infty(M)\) which inherits the ghost number grading as well as the \({\mathbb{Z}}_2\)-grading in even and odd elements. A one-parameter family of formal associative deformations \(\star_\kappa\) of the classical extended super Poisson algebra is defined on \({\mathcal A}[[\lambda]]\), for which the corresponding BRST charge \(\Theta_\kappa\) has square zero. The deformations are equivalent by explicit equivalence transformations \(S_\kappa\). Among the family of quantum BRST operators \({\mathcal D}_\kappa\) in \(({\mathcal A}[[\lambda]], \star_\kappa)\), i.e., the graded star product commutators with \(\Theta_\kappa\), the most natural from the point of Clifford algebras is the Weyl-ordered BRST operator \({\mathcal D}_W=\frac{1}{\text{i} \lambda}\text{ad}_W(\Theta_W)\) corresponding to \(\kappa=1/2\). Here \(\Theta_W=\Omega+{\mathbf J}\), where \(\Omega\) represents the Lie bracket in \({\mathfrak g}\) and \(\text{ad}_W\) is the supercommutator for \(\star_{1/2}\). Under the equivalence transformations \(S_\kappa\), the BRST charge and the BRST operator transform according to \({\mathcal D}_\kappa=S^{-1}_{\kappa-\frac{1}{2}}\circ{\mathcal D}_W\circ S_{\kappa-\frac{1}{2}}\), \(\Theta_\kappa=S^{-1}_{\kappa-\frac{1}{2}}(\Theta_W)\). It turns out that \({\mathcal D}_W\) is not very encouraging concerning cohomology computations, but luckily the quantum BRST operator \({\mathcal D}_0\), associated with the standard ordering, defines a double complex thus giving rise to a deformed version of the classical Koszul boundary operator and the classical Chevalley-Eilenberg differential. For every regular level-zero surface \(C\) of \(J\) one can compute the quantum BRST cohomology of \(({\mathcal A}[[\lambda]],{\mathcal D}_0)\) by means of a deformed augmentation, i.e., a quantized version of restricting functions on \(M\) to \(C\). The result is that the quantum BRST cohomology is isomorphic to the Chevalley-Eilenberg cohomology of the Lie algebra \({\mathfrak g}\) with a \({\mathfrak g}\)-module \(C^\infty(C)[[\lambda]]\). Moreover, the quantum BRST cohomologies of the operators \({\mathcal D}_\kappa\) are all isomorphic. Finally, a deformed version \({\mathbf{\mathcal I}}_C\) of the vanishing ideal of the constrained surface is defined, so that its idealiser \({\mathbf{\mathcal B}}_C\) in \((C^\infty(M)[[\lambda]],*)\) modulo \({\mathbf{\mathcal I}}_C\) turns out to be naturally isomorphic to the quantum BRST algebra. Simple examples and counterexamples are also discussed.


53D55 Deformation quantization, star products
81T70 Quantization in field theory; cohomological methods
53D20 Momentum maps; symplectic reduction
17B56 Cohomology of Lie (super)algebras
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